 Input distribution optimization for a binary symmetric channel

Let $q=\mathrm{Pr}(X=1)$ for the input bit of a binary symmetric channel with crossover probability $p$. The law of total probability givesConsequently its mutual information isHere $p$ specifies the noise and $q$ specifies the input bias; they are different choices. At $p=0.1$ and $q=0.2$, the output has $\mathrm{Pr}(Y=1)=0.26$, giving $H(Y)\approx 0.82674637$ and $I(X;Y)\approx 0.35775078$ bits. With the same noise and the fair input $q=1/2$, the output becomes fair and $I(X;Y)\approx 0.53100441$ bits, attaining binary symmetric channel capacity.

For $p\ne 1/2$, binary entropy reaches one only at output mass $1/2$, and the equation $p+(1-2p)q=1/2$ forces $q=1/2$. Thus the maximizing input is unique in those cases. For $p=1/2$, every input produces the same fair output and zero mutual information, so every input bias is optimal and none transmits information. A fixed input $q=0$ or $q=1$ also carries zero mutual information for every $p$: noise can make its output unpredictable without creating an unknown message to recover.